zero of polynomial

Let R be a subring of a commutative ring S.  If f is a polynomialPlanetmathPlanetmath in R[X], it defines an evaluation homomorphism from S to S.  Any element α of S satisfying

f(α)= 0

is a zero of the polynomial f.

If R also is equipped with a non-zero unity, then the polynomial f is in S[X] divisible by the binomial  X-α (cf. the factor theorem).  In this case, if f is divisible by (X-α)n but not by (X-α)n+1, then α is a zero of the order n of the polynomial f.  If this order is 1, then α is a simple zero of f.

For example, the real number 2 () is a zero of the polynomial X2-2 of the polynomial ring [X].

Title zero of polynomial
Canonical name ZeroOfPolynomial
Date of creation 2013-03-22 18:19:50
Last modified on 2013-03-22 18:19:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 13P05
Classification msc 11C08
Classification msc 12E05
Related topic PolynomialFunction
Related topic ZerosAndPolesOfRationalFunction
Defines zero of polynomial
Defines order of zero
Defines order
Defines simple zero
Defines simple